It depends on how you originally created the artwork. If it was created as vector artwork then you can export it as vector artwork (see Gramps' references) but if it was created as raster artwork you will have to somehow create the vector artwork information.
Without going too much into it, raster artwork is a pixel-by-pixel description of artwork. This is great when you're working with a wide range of colors and photorealistic images (such as photographs) but it has issues when you try to scale it. Vector artwork, however, is a series of points and lines within a space (think string connected to nails pounded into a board - if you switch to a larger board and make sure the nails are pounded into the same relative positions in the new board when you run your string around your nails you'll end up with a larger version of the same image you had before). The down side to this is that if you want photorealistic effects such as gradation you need to describe the gradation across whatever shape you've lined out, which is generally a lot more work.
If you need to convert raster graphics to vector graphics many people start with importing the artwork into Illustrator and using the Live Trace function, then adjusting as needed. It's notoriously "minute to learn, lifetime to master", however, and oftentimes there's a lot of prep involved on the Photoshop side.
As far as file formats, Illustrator is able to save and export as most common filetypes based on need. Illustrator and SVG, for example, contain vector information while .png and .jpg, being raster formats, won't.
TL;DR: Yes, with an "if"; no, with a "but"